| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cardid2 |
|- ( A e. dom card -> ( card ` A ) ~~ A ) |
| 2 |
1
|
ensymd |
|- ( A e. dom card -> A ~~ ( card ` A ) ) |
| 3 |
|
entr |
|- ( ( y ~~ A /\ A ~~ ( card ` A ) ) -> y ~~ ( card ` A ) ) |
| 4 |
3
|
expcom |
|- ( A ~~ ( card ` A ) -> ( y ~~ A -> y ~~ ( card ` A ) ) ) |
| 5 |
2 4
|
syl |
|- ( A e. dom card -> ( y ~~ A -> y ~~ ( card ` A ) ) ) |
| 6 |
|
entr |
|- ( ( y ~~ ( card ` A ) /\ ( card ` A ) ~~ A ) -> y ~~ A ) |
| 7 |
6
|
expcom |
|- ( ( card ` A ) ~~ A -> ( y ~~ ( card ` A ) -> y ~~ A ) ) |
| 8 |
1 7
|
syl |
|- ( A e. dom card -> ( y ~~ ( card ` A ) -> y ~~ A ) ) |
| 9 |
5 8
|
impbid |
|- ( A e. dom card -> ( y ~~ A <-> y ~~ ( card ` A ) ) ) |
| 10 |
9
|
rabbidv |
|- ( A e. dom card -> { y e. On | y ~~ A } = { y e. On | y ~~ ( card ` A ) } ) |
| 11 |
10
|
inteqd |
|- ( A e. dom card -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ ( card ` A ) } ) |
| 12 |
|
cardval3 |
|- ( A e. dom card -> ( card ` A ) = |^| { y e. On | y ~~ A } ) |
| 13 |
|
cardon |
|- ( card ` A ) e. On |
| 14 |
|
oncardval |
|- ( ( card ` A ) e. On -> ( card ` ( card ` A ) ) = |^| { y e. On | y ~~ ( card ` A ) } ) |
| 15 |
13 14
|
mp1i |
|- ( A e. dom card -> ( card ` ( card ` A ) ) = |^| { y e. On | y ~~ ( card ` A ) } ) |
| 16 |
11 12 15
|
3eqtr4rd |
|- ( A e. dom card -> ( card ` ( card ` A ) ) = ( card ` A ) ) |
| 17 |
|
card0 |
|- ( card ` (/) ) = (/) |
| 18 |
|
ndmfv |
|- ( -. A e. dom card -> ( card ` A ) = (/) ) |
| 19 |
18
|
fveq2d |
|- ( -. A e. dom card -> ( card ` ( card ` A ) ) = ( card ` (/) ) ) |
| 20 |
17 19 18
|
3eqtr4a |
|- ( -. A e. dom card -> ( card ` ( card ` A ) ) = ( card ` A ) ) |
| 21 |
16 20
|
pm2.61i |
|- ( card ` ( card ` A ) ) = ( card ` A ) |